Being exothermic does not mean the reaction will be slowed down by heat. Fire, for example, is exothermic and speeds up at higher temperatures.
For rusting, another important factor must be considered: Water.
Iron rusts in the presence of water. Dry iron does not rust.This follows from the chemical reactions of rusting - they require hydrogen atoms.
Salt water is even better because it provides excellent electron transport. But that's not the case here.
The question is: Scalding hot or cold bbq in the rain, which one rusts?
To answer that you need to answer some assumptions.
Are you assuming that the heat burns off the protective layer of the bbq? Are you in fact talking about iron here?Will both bbq's stay in the rain for days? Will the hot one cool down?
I'd say: If the rain is like a fine drizzle which lasts minutes, the cold one will be wet and stay wet - it will rust, while the hot one dries up and stays dry - meaning it won't rust.
If it pours, and both get the same amount of 'wetting', the oxidation will occur faster at the hot one - in that case there is so much water that the temperature cannot boil the water off, it just helps to speed up the reaction.
Of course, if it pours, you have to consider how long the scalding hot bbq will stay hot.